Diversity in Rankingvia Resistive Graph Centers

Avinava DubeyIBM Research India

Soumen ChakrabartiIIT Bombay

Chiranjib BhattacharyyaIISc Bangalore

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PageRank: Conventional view Inputs

• Graph with edge conductance matrix C• Personalized teleport distribution r• Walk with probability, teleport w.p. 1• “Biased random surfer”

Output• Steady state visit distribution• “You should emulate the

aggregate behavior ofmany random surfers” r

i

j

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User view: Exact opposite! Random search-guided surfer Search engine knows relevant subgraph But user can inspect only a few hits Search engine outputs sparse teleport r

Corpus

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User view: Exact opposite! User diffuses out through sparse teleport Occasionally teleports back to search results Eventually explores green subgraph (Red, green “boundaries” are probabilistic)

Corpus

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Diffusion defined via subsumption Original PageRank: diffusion via hyperlinks But frequently used with other kinds of edges Suppose surfer is on page i And, having read i, there is no new info in j Then let C(j|i), also written as C(ij) be

large

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Graph center diversity (GCD) Suppose the searcher can click through at most

three links returned by the search engine If any of the pages could be potentially relevant, … … then we cannot waste teleports on one cluster

A natural definition of diversity

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Formulation summary thus far Search engine knows what’s best for query

• Node i has relevance b(i)

User has limited patience scanning results• r must be sparse: at most K positive elements

Conductance matrix C and walk probability predict user behavior once given r

Steady state visit probabilities given by

Inference, hard: design sparse r to minimize

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Attention decay profile

Design a teleport r with decaying weights

So as to align weighted merged clouds with b

Attentionprofile

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Learning subsumption C(ij) How strongly does i render j redundant? Associate edge ij with features { f } Each f has associated fixed conductance

matrix Cf and personalized PageRanks Mf

Training: Given diverse node sets (r*), learn the convex combination defined by

Simple heuristic (convex optimization):

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Structured learning style formulation More accurately, any r r* should do

worse Define a loss

Combine over query instances

Paper gives an online update algorithm to improve iteratively (exponentiated gradient)

Divergence for r Divergence for r*

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Marginal utility methods Max marginal relevance (MMR)

• Given q, already chose subset S; next choice is

SubTopic• Similar to MMR

• sim1 and sim2 use probabilistic topic models

SVMdiv• Learns subtopic coverage from word coverage

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PageRank based diversity models Grasshopper

• Edges associated with fixed similarity scores• Best node has highest PageRank• Make best node a sink, run PageRank again• Note, no meaningful steady state, Pr(sink)=1• Next best node has largest expected number of

visits before walk absorbed in sink

DivRank• With visits to node j, inbound edges get thicker• Rich gets even richer than you expected• Tiebreaking causes one cluster member to win

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Submodular set selection Sounds similar to MMR but on a graph

• Undirected edge (i,j) has weight wij

Given node set V, select subset S so as to• Maximize coverage of V \S:• Minimize redundancy within S:

Additional size budget constraint Hard, but provable approximations No learning of edge weight/conductance

S V \S

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Experiments: Three diverse domains Subtopic information retrieval (TREC)

• Query under-specified or ambiguous• Balance responses across subtopics or facets

Social network search (IMDB)• List high-prestige actors without knowing country• Diversity many countries covered

Extractive document summarization (DUC)• Choose subset of sentences• That are representative of the whole document• And do not render each other redundant

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Subtopic information retrieval results

Ground truth has subtopics covered by each doc Subtopic-aware precision vs. recall GCD dominates other subtopic IR approaches

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Effect of training

Uniform all f equal Maxent = convex heuristic minimizing KL

divergence between b and PageRank EG = Exponentiated gradient Successive improvements in subtopic-aware mean

average precision

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Ranking in social networks (IMDB) 3452 actors, 1027

movies, 47 countries Actor’s prestige

depends on prestigeof movies where s/he has worked

Rank actors by prestige

GCD rapidly increases distinct countries

While also increasing number of movies

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Document summarization DUC 2004, task 2, ROUGE-1 30, 20 summaries to train, test MMR, SubTopic not

competitive Associative graph diffusion

(Grasshopper, DivRank) worse than GCD and Submodular

GCD comparable to Submodular even without using sentence size budget constraints

Algorithm Train Test

MMR 0.324 0.32

SubTopic 0.32 0.323

Grasshopper 0.341 0.33

DivRank 0.353 0.345

GCD 0.377 0.374

Submodular 0.389 0.373

Optimal 0.421 0.407

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Conclusion A novel model for redundancy and diversity Based on an “inverted” notion of PageRank Inference amounts to finding centers in

conductance graphs• “GCD”, graph center diversity

Bonus: learn conductance via edge features GCD shows better or similar performance in

three diverse application domains

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